Properties

Label 1216.2.i.k.961.1
Level $1216$
Weight $2$
Character 1216.961
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(577,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(-1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 1216.961
Dual form 1216.2.i.k.577.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.32288 + 2.29129i) q^{3} +(-0.822876 + 1.42526i) q^{5} -3.64575 q^{7} +(-2.00000 - 3.46410i) q^{9} +O(q^{10})\) \(q+(-1.32288 + 2.29129i) q^{3} +(-0.822876 + 1.42526i) q^{5} -3.64575 q^{7} +(-2.00000 - 3.46410i) q^{9} -4.64575 q^{11} +(1.00000 + 1.73205i) q^{13} +(-2.17712 - 3.77089i) q^{15} +(1.67712 - 4.02334i) q^{19} +(4.82288 - 8.35347i) q^{21} +(-0.822876 - 1.42526i) q^{23} +(1.14575 + 1.98450i) q^{25} +2.64575 q^{27} +(0.822876 + 1.42526i) q^{29} +5.64575 q^{31} +(6.14575 - 10.6448i) q^{33} +(3.00000 - 5.19615i) q^{35} -0.354249 q^{37} -5.29150 q^{39} +(0.145751 - 0.252449i) q^{41} +(-5.64575 + 9.77873i) q^{43} +6.58301 q^{45} +(2.17712 + 3.77089i) q^{47} +6.29150 q^{49} +(-6.29150 - 10.8972i) q^{53} +(3.82288 - 6.62141i) q^{55} +(7.00000 + 9.16515i) q^{57} +(3.96863 - 6.87386i) q^{59} +(0.468627 + 0.811686i) q^{61} +(7.29150 + 12.6293i) q^{63} -3.29150 q^{65} +(-0.322876 - 0.559237i) q^{67} +4.35425 q^{69} +(-1.35425 + 2.34563i) q^{71} +(-0.854249 + 1.47960i) q^{73} -6.06275 q^{75} +16.9373 q^{77} +(-2.00000 + 3.46410i) q^{79} +(2.50000 - 4.33013i) q^{81} +7.93725 q^{83} -4.35425 q^{87} +(-3.64575 - 6.31463i) q^{91} +(-7.46863 + 12.9360i) q^{93} +(4.35425 + 5.70105i) q^{95} +(1.85425 - 3.21165i) q^{97} +(9.29150 + 16.0934i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 4 q^{7} - 8 q^{9} - 8 q^{11} + 4 q^{13} - 14 q^{15} + 12 q^{19} + 14 q^{21} + 2 q^{23} - 6 q^{25} - 2 q^{29} + 12 q^{31} + 14 q^{33} + 12 q^{35} - 12 q^{37} - 10 q^{41} - 12 q^{43} - 16 q^{45} + 14 q^{47} + 4 q^{49} - 4 q^{53} + 10 q^{55} + 28 q^{57} - 14 q^{61} + 8 q^{63} + 8 q^{65} + 4 q^{67} + 28 q^{69} - 16 q^{71} - 14 q^{73} - 56 q^{75} + 36 q^{77} - 8 q^{79} + 10 q^{81} - 28 q^{87} - 4 q^{91} - 14 q^{93} + 28 q^{95} + 18 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.32288 + 2.29129i −0.763763 + 1.32288i 0.177136 + 0.984186i \(0.443317\pi\)
−0.940898 + 0.338689i \(0.890016\pi\)
\(4\) 0 0
\(5\) −0.822876 + 1.42526i −0.368001 + 0.637397i −0.989253 0.146214i \(-0.953291\pi\)
0.621252 + 0.783611i \(0.286624\pi\)
\(6\) 0 0
\(7\) −3.64575 −1.37796 −0.688982 0.724778i \(-0.741942\pi\)
−0.688982 + 0.724778i \(0.741942\pi\)
\(8\) 0 0
\(9\) −2.00000 3.46410i −0.666667 1.15470i
\(10\) 0 0
\(11\) −4.64575 −1.40075 −0.700373 0.713777i \(-0.746983\pi\)
−0.700373 + 0.713777i \(0.746983\pi\)
\(12\) 0 0
\(13\) 1.00000 + 1.73205i 0.277350 + 0.480384i 0.970725 0.240192i \(-0.0772105\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) −2.17712 3.77089i −0.562131 0.973640i
\(16\) 0 0
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 1.67712 4.02334i 0.384759 0.923017i
\(20\) 0 0
\(21\) 4.82288 8.35347i 1.05244 1.82288i
\(22\) 0 0
\(23\) −0.822876 1.42526i −0.171581 0.297188i 0.767391 0.641179i \(-0.221554\pi\)
−0.938973 + 0.343991i \(0.888221\pi\)
\(24\) 0 0
\(25\) 1.14575 + 1.98450i 0.229150 + 0.396900i
\(26\) 0 0
\(27\) 2.64575 0.509175
\(28\) 0 0
\(29\) 0.822876 + 1.42526i 0.152804 + 0.264665i 0.932257 0.361796i \(-0.117836\pi\)
−0.779453 + 0.626461i \(0.784503\pi\)
\(30\) 0 0
\(31\) 5.64575 1.01401 0.507003 0.861944i \(-0.330753\pi\)
0.507003 + 0.861944i \(0.330753\pi\)
\(32\) 0 0
\(33\) 6.14575 10.6448i 1.06984 1.85301i
\(34\) 0 0
\(35\) 3.00000 5.19615i 0.507093 0.878310i
\(36\) 0 0
\(37\) −0.354249 −0.0582381 −0.0291191 0.999576i \(-0.509270\pi\)
−0.0291191 + 0.999576i \(0.509270\pi\)
\(38\) 0 0
\(39\) −5.29150 −0.847319
\(40\) 0 0
\(41\) 0.145751 0.252449i 0.0227625 0.0394259i −0.854420 0.519583i \(-0.826087\pi\)
0.877182 + 0.480158i \(0.159421\pi\)
\(42\) 0 0
\(43\) −5.64575 + 9.77873i −0.860969 + 1.49124i 0.0100257 + 0.999950i \(0.496809\pi\)
−0.870995 + 0.491292i \(0.836525\pi\)
\(44\) 0 0
\(45\) 6.58301 0.981336
\(46\) 0 0
\(47\) 2.17712 + 3.77089i 0.317566 + 0.550041i 0.979980 0.199098i \(-0.0638011\pi\)
−0.662413 + 0.749138i \(0.730468\pi\)
\(48\) 0 0
\(49\) 6.29150 0.898786
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.29150 10.8972i −0.864204 1.49685i −0.867835 0.496852i \(-0.834489\pi\)
0.00363070 0.999993i \(-0.498844\pi\)
\(54\) 0 0
\(55\) 3.82288 6.62141i 0.515476 0.892831i
\(56\) 0 0
\(57\) 7.00000 + 9.16515i 0.927173 + 1.21395i
\(58\) 0 0
\(59\) 3.96863 6.87386i 0.516671 0.894901i −0.483141 0.875542i \(-0.660504\pi\)
0.999813 0.0193585i \(-0.00616237\pi\)
\(60\) 0 0
\(61\) 0.468627 + 0.811686i 0.0600015 + 0.103926i 0.894466 0.447136i \(-0.147556\pi\)
−0.834464 + 0.551062i \(0.814223\pi\)
\(62\) 0 0
\(63\) 7.29150 + 12.6293i 0.918643 + 1.59114i
\(64\) 0 0
\(65\) −3.29150 −0.408261
\(66\) 0 0
\(67\) −0.322876 0.559237i −0.0394455 0.0683217i 0.845629 0.533772i \(-0.179226\pi\)
−0.885074 + 0.465450i \(0.845892\pi\)
\(68\) 0 0
\(69\) 4.35425 0.524190
\(70\) 0 0
\(71\) −1.35425 + 2.34563i −0.160720 + 0.278375i −0.935127 0.354313i \(-0.884715\pi\)
0.774407 + 0.632687i \(0.218048\pi\)
\(72\) 0 0
\(73\) −0.854249 + 1.47960i −0.0999822 + 0.173174i −0.911677 0.410907i \(-0.865212\pi\)
0.811695 + 0.584082i \(0.198545\pi\)
\(74\) 0 0
\(75\) −6.06275 −0.700066
\(76\) 0 0
\(77\) 16.9373 1.93018
\(78\) 0 0
\(79\) −2.00000 + 3.46410i −0.225018 + 0.389742i −0.956325 0.292306i \(-0.905577\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 2.50000 4.33013i 0.277778 0.481125i
\(82\) 0 0
\(83\) 7.93725 0.871227 0.435613 0.900134i \(-0.356531\pi\)
0.435613 + 0.900134i \(0.356531\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.35425 −0.466824
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) −3.64575 6.31463i −0.382179 0.661953i
\(92\) 0 0
\(93\) −7.46863 + 12.9360i −0.774461 + 1.34141i
\(94\) 0 0
\(95\) 4.35425 + 5.70105i 0.446736 + 0.584915i
\(96\) 0 0
\(97\) 1.85425 3.21165i 0.188270 0.326094i −0.756403 0.654106i \(-0.773045\pi\)
0.944674 + 0.328012i \(0.106379\pi\)
\(98\) 0 0
\(99\) 9.29150 + 16.0934i 0.933831 + 1.61744i
\(100\) 0 0
\(101\) −6.82288 11.8176i −0.678902 1.17589i −0.975312 0.220831i \(-0.929123\pi\)
0.296411 0.955061i \(-0.404210\pi\)
\(102\) 0 0
\(103\) 13.2915 1.30965 0.654825 0.755780i \(-0.272742\pi\)
0.654825 + 0.755780i \(0.272742\pi\)
\(104\) 0 0
\(105\) 7.93725 + 13.7477i 0.774597 + 1.34164i
\(106\) 0 0
\(107\) 15.2915 1.47829 0.739143 0.673549i \(-0.235231\pi\)
0.739143 + 0.673549i \(0.235231\pi\)
\(108\) 0 0
\(109\) 7.29150 12.6293i 0.698399 1.20966i −0.270622 0.962686i \(-0.587229\pi\)
0.969021 0.246977i \(-0.0794373\pi\)
\(110\) 0 0
\(111\) 0.468627 0.811686i 0.0444801 0.0770418i
\(112\) 0 0
\(113\) −15.5830 −1.46593 −0.732963 0.680269i \(-0.761863\pi\)
−0.732963 + 0.680269i \(0.761863\pi\)
\(114\) 0 0
\(115\) 2.70850 0.252569
\(116\) 0 0
\(117\) 4.00000 6.92820i 0.369800 0.640513i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.5830 0.962091
\(122\) 0 0
\(123\) 0.385622 + 0.667916i 0.0347703 + 0.0602240i
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −6.64575 11.5108i −0.589715 1.02142i −0.994270 0.106903i \(-0.965907\pi\)
0.404554 0.914514i \(-0.367427\pi\)
\(128\) 0 0
\(129\) −14.9373 25.8721i −1.31515 2.27791i
\(130\) 0 0
\(131\) −0.968627 + 1.67771i −0.0846293 + 0.146582i −0.905233 0.424915i \(-0.860304\pi\)
0.820604 + 0.571497i \(0.193637\pi\)
\(132\) 0 0
\(133\) −6.11438 + 14.6681i −0.530184 + 1.27188i
\(134\) 0 0
\(135\) −2.17712 + 3.77089i −0.187377 + 0.324547i
\(136\) 0 0
\(137\) −7.79150 13.4953i −0.665673 1.15298i −0.979102 0.203368i \(-0.934811\pi\)
0.313429 0.949612i \(-0.398522\pi\)
\(138\) 0 0
\(139\) −9.32288 16.1477i −0.790756 1.36963i −0.925499 0.378749i \(-0.876354\pi\)
0.134743 0.990881i \(-0.456979\pi\)
\(140\) 0 0
\(141\) −11.5203 −0.970181
\(142\) 0 0
\(143\) −4.64575 8.04668i −0.388497 0.672897i
\(144\) 0 0
\(145\) −2.70850 −0.224928
\(146\) 0 0
\(147\) −8.32288 + 14.4156i −0.686459 + 1.18898i
\(148\) 0 0
\(149\) 5.46863 9.47194i 0.448007 0.775972i −0.550249 0.835001i \(-0.685467\pi\)
0.998256 + 0.0590292i \(0.0188005\pi\)
\(150\) 0 0
\(151\) −12.9373 −1.05282 −0.526409 0.850231i \(-0.676462\pi\)
−0.526409 + 0.850231i \(0.676462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.64575 + 8.04668i −0.373156 + 0.646325i
\(156\) 0 0
\(157\) −5.29150 + 9.16515i −0.422308 + 0.731459i −0.996165 0.0874969i \(-0.972113\pi\)
0.573857 + 0.818956i \(0.305447\pi\)
\(158\) 0 0
\(159\) 33.2915 2.64019
\(160\) 0 0
\(161\) 3.00000 + 5.19615i 0.236433 + 0.409514i
\(162\) 0 0
\(163\) 3.93725 0.308390 0.154195 0.988040i \(-0.450722\pi\)
0.154195 + 0.988040i \(0.450722\pi\)
\(164\) 0 0
\(165\) 10.1144 + 17.5186i 0.787403 + 1.36382i
\(166\) 0 0
\(167\) −6.00000 10.3923i −0.464294 0.804181i 0.534875 0.844931i \(-0.320359\pi\)
−0.999169 + 0.0407502i \(0.987025\pi\)
\(168\) 0 0
\(169\) 4.50000 7.79423i 0.346154 0.599556i
\(170\) 0 0
\(171\) −17.2915 + 2.23695i −1.32231 + 0.171064i
\(172\) 0 0
\(173\) −3.00000 + 5.19615i −0.228086 + 0.395056i −0.957241 0.289292i \(-0.906580\pi\)
0.729155 + 0.684349i \(0.239913\pi\)
\(174\) 0 0
\(175\) −4.17712 7.23499i −0.315761 0.546914i
\(176\) 0 0
\(177\) 10.5000 + 18.1865i 0.789228 + 1.36698i
\(178\) 0 0
\(179\) 4.06275 0.303664 0.151832 0.988406i \(-0.451483\pi\)
0.151832 + 0.988406i \(0.451483\pi\)
\(180\) 0 0
\(181\) 11.1144 + 19.2507i 0.826125 + 1.43089i 0.901056 + 0.433702i \(0.142793\pi\)
−0.0749311 + 0.997189i \(0.523874\pi\)
\(182\) 0 0
\(183\) −2.47974 −0.183308
\(184\) 0 0
\(185\) 0.291503 0.504897i 0.0214317 0.0371208i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −9.64575 −0.701625
\(190\) 0 0
\(191\) −6.58301 −0.476330 −0.238165 0.971225i \(-0.576546\pi\)
−0.238165 + 0.971225i \(0.576546\pi\)
\(192\) 0 0
\(193\) −7.29150 + 12.6293i −0.524854 + 0.909074i 0.474727 + 0.880133i \(0.342547\pi\)
−0.999581 + 0.0289406i \(0.990787\pi\)
\(194\) 0 0
\(195\) 4.35425 7.54178i 0.311814 0.540078i
\(196\) 0 0
\(197\) 7.64575 0.544737 0.272369 0.962193i \(-0.412193\pi\)
0.272369 + 0.962193i \(0.412193\pi\)
\(198\) 0 0
\(199\) −9.93725 17.2118i −0.704433 1.22011i −0.966896 0.255172i \(-0.917868\pi\)
0.262462 0.964942i \(-0.415465\pi\)
\(200\) 0 0
\(201\) 1.70850 0.120508
\(202\) 0 0
\(203\) −3.00000 5.19615i −0.210559 0.364698i
\(204\) 0 0
\(205\) 0.239870 + 0.415468i 0.0167533 + 0.0290175i
\(206\) 0 0
\(207\) −3.29150 + 5.70105i −0.228775 + 0.396250i
\(208\) 0 0
\(209\) −7.79150 + 18.6914i −0.538950 + 1.29291i
\(210\) 0 0
\(211\) 6.64575 11.5108i 0.457512 0.792435i −0.541316 0.840819i \(-0.682074\pi\)
0.998829 + 0.0483843i \(0.0154072\pi\)
\(212\) 0 0
\(213\) −3.58301 6.20595i −0.245503 0.425224i
\(214\) 0 0
\(215\) −9.29150 16.0934i −0.633675 1.09756i
\(216\) 0 0
\(217\) −20.5830 −1.39727
\(218\) 0 0
\(219\) −2.26013 3.91466i −0.152725 0.264528i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −9.40588 + 16.2915i −0.629864 + 1.09096i 0.357714 + 0.933831i \(0.383556\pi\)
−0.987579 + 0.157126i \(0.949777\pi\)
\(224\) 0 0
\(225\) 4.58301 7.93800i 0.305534 0.529200i
\(226\) 0 0
\(227\) −7.35425 −0.488119 −0.244059 0.969760i \(-0.578479\pi\)
−0.244059 + 0.969760i \(0.578479\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) −22.4059 + 38.8081i −1.47420 + 2.55339i
\(232\) 0 0
\(233\) −9.43725 + 16.3458i −0.618255 + 1.07085i 0.371549 + 0.928413i \(0.378827\pi\)
−0.989804 + 0.142436i \(0.954507\pi\)
\(234\) 0 0
\(235\) −7.16601 −0.467459
\(236\) 0 0
\(237\) −5.29150 9.16515i −0.343720 0.595341i
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 3.79150 + 6.56708i 0.244232 + 0.423022i 0.961915 0.273347i \(-0.0881308\pi\)
−0.717683 + 0.696370i \(0.754797\pi\)
\(242\) 0 0
\(243\) 10.5830 + 18.3303i 0.678900 + 1.17589i
\(244\) 0 0
\(245\) −5.17712 + 8.96704i −0.330754 + 0.572883i
\(246\) 0 0
\(247\) 8.64575 1.11847i 0.550116 0.0711668i
\(248\) 0 0
\(249\) −10.5000 + 18.1865i −0.665410 + 1.15252i
\(250\) 0 0
\(251\) −14.6144 25.3128i −0.922451 1.59773i −0.795610 0.605810i \(-0.792849\pi\)
−0.126842 0.991923i \(-0.540484\pi\)
\(252\) 0 0
\(253\) 3.82288 + 6.62141i 0.240342 + 0.416285i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.14575 10.6448i −0.383361 0.664001i 0.608179 0.793800i \(-0.291900\pi\)
−0.991540 + 0.129798i \(0.958567\pi\)
\(258\) 0 0
\(259\) 1.29150 0.0802501
\(260\) 0 0
\(261\) 3.29150 5.70105i 0.203739 0.352886i
\(262\) 0 0
\(263\) −5.46863 + 9.47194i −0.337210 + 0.584065i −0.983907 0.178682i \(-0.942817\pi\)
0.646697 + 0.762747i \(0.276150\pi\)
\(264\) 0 0
\(265\) 20.7085 1.27211
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 6.17712 10.6991i 0.375234 0.649924i −0.615128 0.788427i \(-0.710896\pi\)
0.990362 + 0.138503i \(0.0442292\pi\)
\(272\) 0 0
\(273\) 19.2915 1.16757
\(274\) 0 0
\(275\) −5.32288 9.21949i −0.320981 0.555956i
\(276\) 0 0
\(277\) 27.5203 1.65353 0.826766 0.562546i \(-0.190178\pi\)
0.826766 + 0.562546i \(0.190178\pi\)
\(278\) 0 0
\(279\) −11.2915 19.5575i −0.676005 1.17087i
\(280\) 0 0
\(281\) −12.7288 22.0469i −0.759334 1.31520i −0.943191 0.332252i \(-0.892192\pi\)
0.183857 0.982953i \(-0.441142\pi\)
\(282\) 0 0
\(283\) −15.3229 + 26.5400i −0.910850 + 1.57764i −0.0979848 + 0.995188i \(0.531240\pi\)
−0.812866 + 0.582451i \(0.802094\pi\)
\(284\) 0 0
\(285\) −18.8229 + 2.43506i −1.11497 + 0.144240i
\(286\) 0 0
\(287\) −0.531373 + 0.920365i −0.0313660 + 0.0543274i
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 4.90588 + 8.49723i 0.287588 + 0.498117i
\(292\) 0 0
\(293\) −28.9373 −1.69053 −0.845266 0.534345i \(-0.820558\pi\)
−0.845266 + 0.534345i \(0.820558\pi\)
\(294\) 0 0
\(295\) 6.53137 + 11.3127i 0.380271 + 0.658649i
\(296\) 0 0
\(297\) −12.2915 −0.713225
\(298\) 0 0
\(299\) 1.64575 2.85052i 0.0951763 0.164850i
\(300\) 0 0
\(301\) 20.5830 35.6508i 1.18638 2.05488i
\(302\) 0 0
\(303\) 36.1033 2.07408
\(304\) 0 0
\(305\) −1.54249 −0.0883225
\(306\) 0 0
\(307\) −0.322876 + 0.559237i −0.0184275 + 0.0319173i −0.875092 0.483956i \(-0.839199\pi\)
0.856665 + 0.515874i \(0.172533\pi\)
\(308\) 0 0
\(309\) −17.5830 + 30.4547i −1.00026 + 1.73250i
\(310\) 0 0
\(311\) −13.6458 −0.773780 −0.386890 0.922126i \(-0.626451\pi\)
−0.386890 + 0.922126i \(0.626451\pi\)
\(312\) 0 0
\(313\) −4.43725 7.68555i −0.250808 0.434413i 0.712940 0.701225i \(-0.247363\pi\)
−0.963749 + 0.266812i \(0.914030\pi\)
\(314\) 0 0
\(315\) −24.0000 −1.35225
\(316\) 0 0
\(317\) −3.00000 5.19615i −0.168497 0.291845i 0.769395 0.638774i \(-0.220558\pi\)
−0.937892 + 0.346929i \(0.887225\pi\)
\(318\) 0 0
\(319\) −3.82288 6.62141i −0.214040 0.370728i
\(320\) 0 0
\(321\) −20.2288 + 35.0372i −1.12906 + 1.95559i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.29150 + 3.96900i −0.127110 + 0.220160i
\(326\) 0 0
\(327\) 19.2915 + 33.4139i 1.06682 + 1.84779i
\(328\) 0 0
\(329\) −7.93725 13.7477i −0.437595 0.757937i
\(330\) 0 0
\(331\) 19.8118 1.08895 0.544476 0.838776i \(-0.316728\pi\)
0.544476 + 0.838776i \(0.316728\pi\)
\(332\) 0 0
\(333\) 0.708497 + 1.22715i 0.0388254 + 0.0672476i
\(334\) 0 0
\(335\) 1.06275 0.0580640
\(336\) 0 0
\(337\) 4.85425 8.40781i 0.264428 0.458002i −0.702986 0.711204i \(-0.748150\pi\)
0.967414 + 0.253202i \(0.0814836\pi\)
\(338\) 0 0
\(339\) 20.6144 35.7052i 1.11962 1.93924i
\(340\) 0 0
\(341\) −26.2288 −1.42037
\(342\) 0 0
\(343\) 2.58301 0.139469
\(344\) 0 0
\(345\) −3.58301 + 6.20595i −0.192903 + 0.334117i
\(346\) 0 0
\(347\) 11.6144 20.1167i 0.623492 1.07992i −0.365338 0.930875i \(-0.619047\pi\)
0.988830 0.149046i \(-0.0476201\pi\)
\(348\) 0 0
\(349\) −21.1660 −1.13299 −0.566495 0.824065i \(-0.691701\pi\)
−0.566495 + 0.824065i \(0.691701\pi\)
\(350\) 0 0
\(351\) 2.64575 + 4.58258i 0.141220 + 0.244600i
\(352\) 0 0
\(353\) 12.8745 0.685241 0.342620 0.939474i \(-0.388686\pi\)
0.342620 + 0.939474i \(0.388686\pi\)
\(354\) 0 0
\(355\) −2.22876 3.86032i −0.118290 0.204884i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.46863 + 4.27579i −0.130289 + 0.225667i −0.923788 0.382904i \(-0.874924\pi\)
0.793499 + 0.608572i \(0.208257\pi\)
\(360\) 0 0
\(361\) −13.3745 13.4953i −0.703921 0.710278i
\(362\) 0 0
\(363\) −14.0000 + 24.2487i −0.734809 + 1.27273i
\(364\) 0 0
\(365\) −1.40588 2.43506i −0.0735872 0.127457i
\(366\) 0 0
\(367\) 8.11438 + 14.0545i 0.423567 + 0.733640i 0.996285 0.0861125i \(-0.0274444\pi\)
−0.572718 + 0.819752i \(0.694111\pi\)
\(368\) 0 0
\(369\) −1.16601 −0.0607001
\(370\) 0 0
\(371\) 22.9373 + 39.7285i 1.19084 + 2.06260i
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 15.8745 27.4955i 0.819756 1.41986i
\(376\) 0 0
\(377\) −1.64575 + 2.85052i −0.0847605 + 0.146810i
\(378\) 0 0
\(379\) 10.7085 0.550059 0.275029 0.961436i \(-0.411312\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(380\) 0 0
\(381\) 35.1660 1.80161
\(382\) 0 0
\(383\) 2.76013 4.78068i 0.141036 0.244282i −0.786851 0.617143i \(-0.788290\pi\)
0.927887 + 0.372861i \(0.121623\pi\)
\(384\) 0 0
\(385\) −13.9373 + 24.1400i −0.710308 + 1.23029i
\(386\) 0 0
\(387\) 45.1660 2.29592
\(388\) 0 0
\(389\) 6.00000 + 10.3923i 0.304212 + 0.526911i 0.977086 0.212847i \(-0.0682735\pi\)
−0.672874 + 0.739758i \(0.734940\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −2.56275 4.43881i −0.129273 0.223908i
\(394\) 0 0
\(395\) −3.29150 5.70105i −0.165613 0.286851i
\(396\) 0 0
\(397\) 10.5314 18.2409i 0.528554 0.915483i −0.470891 0.882191i \(-0.656068\pi\)
0.999446 0.0332919i \(-0.0105991\pi\)
\(398\) 0 0
\(399\) −25.5203 33.4139i −1.27761 1.67279i
\(400\) 0 0
\(401\) −13.7915 + 23.8876i −0.688715 + 1.19289i 0.283539 + 0.958961i \(0.408491\pi\)
−0.972254 + 0.233928i \(0.924842\pi\)
\(402\) 0 0
\(403\) 5.64575 + 9.77873i 0.281235 + 0.487113i
\(404\) 0 0
\(405\) 4.11438 + 7.12631i 0.204445 + 0.354109i
\(406\) 0 0
\(407\) 1.64575 0.0815769
\(408\) 0 0
\(409\) 3.79150 + 6.56708i 0.187478 + 0.324721i 0.944409 0.328774i \(-0.106635\pi\)
−0.756931 + 0.653495i \(0.773302\pi\)
\(410\) 0 0
\(411\) 41.2288 2.03366
\(412\) 0 0
\(413\) −14.4686 + 25.0604i −0.711955 + 1.23314i
\(414\) 0 0
\(415\) −6.53137 + 11.3127i −0.320612 + 0.555317i
\(416\) 0 0
\(417\) 49.3320 2.41580
\(418\) 0 0
\(419\) −31.7490 −1.55104 −0.775520 0.631322i \(-0.782512\pi\)
−0.775520 + 0.631322i \(0.782512\pi\)
\(420\) 0 0
\(421\) −12.4059 + 21.4876i −0.604626 + 1.04724i 0.387485 + 0.921876i \(0.373344\pi\)
−0.992111 + 0.125366i \(0.959989\pi\)
\(422\) 0 0
\(423\) 8.70850 15.0836i 0.423422 0.733388i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.70850 2.95920i −0.0826800 0.143206i
\(428\) 0 0
\(429\) 24.5830 1.18688
\(430\) 0 0
\(431\) 13.9373 + 24.1400i 0.671334 + 1.16278i 0.977526 + 0.210815i \(0.0676117\pi\)
−0.306192 + 0.951970i \(0.599055\pi\)
\(432\) 0 0
\(433\) −8.93725 15.4798i −0.429497 0.743911i 0.567332 0.823489i \(-0.307976\pi\)
−0.996829 + 0.0795788i \(0.974642\pi\)
\(434\) 0 0
\(435\) 3.58301 6.20595i 0.171792 0.297552i
\(436\) 0 0
\(437\) −7.11438 + 0.920365i −0.340327 + 0.0440270i
\(438\) 0 0
\(439\) 5.40588 9.36326i 0.258009 0.446884i −0.707700 0.706513i \(-0.750267\pi\)
0.965708 + 0.259629i \(0.0836004\pi\)
\(440\) 0 0
\(441\) −12.5830 21.7944i −0.599191 1.03783i
\(442\) 0 0
\(443\) −5.32288 9.21949i −0.252897 0.438031i 0.711425 0.702762i \(-0.248050\pi\)
−0.964322 + 0.264731i \(0.914717\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 14.4686 + 25.0604i 0.684343 + 1.18532i
\(448\) 0 0
\(449\) −24.2915 −1.14639 −0.573193 0.819420i \(-0.694296\pi\)
−0.573193 + 0.819420i \(0.694296\pi\)
\(450\) 0 0
\(451\) −0.677124 + 1.17281i −0.0318845 + 0.0552256i
\(452\) 0 0
\(453\) 17.1144 29.6430i 0.804104 1.39275i
\(454\) 0 0
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) 32.8745 1.53780 0.768902 0.639366i \(-0.220803\pi\)
0.768902 + 0.639366i \(0.220803\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.58301 16.5983i 0.446325 0.773058i −0.551818 0.833964i \(-0.686066\pi\)
0.998143 + 0.0609066i \(0.0193992\pi\)
\(462\) 0 0
\(463\) 38.4575 1.78727 0.893636 0.448792i \(-0.148146\pi\)
0.893636 + 0.448792i \(0.148146\pi\)
\(464\) 0 0
\(465\) −12.2915 21.2895i −0.570005 0.987277i
\(466\) 0 0
\(467\) −19.3542 −0.895608 −0.447804 0.894132i \(-0.647794\pi\)
−0.447804 + 0.894132i \(0.647794\pi\)
\(468\) 0 0
\(469\) 1.17712 + 2.03884i 0.0543546 + 0.0941448i
\(470\) 0 0
\(471\) −14.0000 24.2487i −0.645086 1.11732i
\(472\) 0 0
\(473\) 26.2288 45.4295i 1.20600 2.08885i
\(474\) 0 0
\(475\) 9.90588 1.28149i 0.454513 0.0587989i
\(476\) 0 0
\(477\) −25.1660 + 43.5888i −1.15227 + 1.99579i
\(478\) 0 0
\(479\) 3.29150 + 5.70105i 0.150393 + 0.260488i 0.931372 0.364069i \(-0.118613\pi\)
−0.780979 + 0.624557i \(0.785280\pi\)
\(480\) 0 0
\(481\) −0.354249 0.613577i −0.0161523 0.0279767i
\(482\) 0 0
\(483\) −15.8745 −0.722315
\(484\) 0 0
\(485\) 3.05163 + 5.28558i 0.138567 + 0.240006i
\(486\) 0 0
\(487\) −4.22876 −0.191623 −0.0958116 0.995399i \(-0.530545\pi\)
−0.0958116 + 0.995399i \(0.530545\pi\)
\(488\) 0 0
\(489\) −5.20850 + 9.02138i −0.235536 + 0.407961i
\(490\) 0 0
\(491\) −19.6458 + 34.0274i −0.886600 + 1.53564i −0.0427320 + 0.999087i \(0.513606\pi\)
−0.843868 + 0.536550i \(0.819727\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −30.5830 −1.37460
\(496\) 0 0
\(497\) 4.93725 8.55157i 0.221466 0.383591i
\(498\) 0 0
\(499\) 2.38562 4.13202i 0.106795 0.184975i −0.807675 0.589628i \(-0.799274\pi\)
0.914470 + 0.404653i \(0.132608\pi\)
\(500\) 0 0
\(501\) 31.7490 1.41844
\(502\) 0 0
\(503\) −20.4686 35.4527i −0.912651 1.58076i −0.810305 0.586009i \(-0.800698\pi\)
−0.102346 0.994749i \(-0.532635\pi\)
\(504\) 0 0
\(505\) 22.4575 0.999346
\(506\) 0 0
\(507\) 11.9059 + 20.6216i 0.528759 + 0.915837i
\(508\) 0 0
\(509\) −15.8745 27.4955i −0.703625 1.21871i −0.967185 0.254072i \(-0.918230\pi\)
0.263560 0.964643i \(-0.415103\pi\)
\(510\) 0 0
\(511\) 3.11438 5.39426i 0.137772 0.238628i
\(512\) 0 0
\(513\) 4.43725 10.6448i 0.195910 0.469977i
\(514\) 0 0
\(515\) −10.9373 + 18.9439i −0.481953 + 0.834767i
\(516\) 0 0
\(517\) −10.1144 17.5186i −0.444830 0.770468i
\(518\) 0 0
\(519\) −7.93725 13.7477i −0.348407 0.603458i
\(520\) 0 0
\(521\) −11.7085 −0.512959 −0.256479 0.966550i \(-0.582563\pi\)
−0.256479 + 0.966550i \(0.582563\pi\)
\(522\) 0 0
\(523\) 0.937254 + 1.62337i 0.0409833 + 0.0709851i 0.885789 0.464087i \(-0.153618\pi\)
−0.844806 + 0.535072i \(0.820284\pi\)
\(524\) 0 0
\(525\) 22.1033 0.964666
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 10.1458 17.5730i 0.441120 0.764042i
\(530\) 0 0
\(531\) −31.7490 −1.37779
\(532\) 0 0
\(533\) 0.583005 0.0252528
\(534\) 0 0
\(535\) −12.5830 + 21.7944i −0.544011 + 0.942254i
\(536\) 0 0
\(537\) −5.37451 + 9.30892i −0.231927 + 0.401710i
\(538\) 0 0
\(539\) −29.2288 −1.25897
\(540\) 0 0
\(541\) 4.00000 + 6.92820i 0.171973 + 0.297867i 0.939110 0.343617i \(-0.111652\pi\)
−0.767136 + 0.641484i \(0.778319\pi\)
\(542\) 0 0
\(543\) −58.8118 −2.52385
\(544\) 0 0
\(545\) 12.0000 + 20.7846i 0.514024 + 0.890315i
\(546\) 0 0
\(547\) −5.64575 9.77873i −0.241395 0.418108i 0.719717 0.694268i \(-0.244272\pi\)
−0.961112 + 0.276159i \(0.910938\pi\)
\(548\) 0 0
\(549\) 1.87451 3.24674i 0.0800020 0.138568i
\(550\) 0 0
\(551\) 7.11438 0.920365i 0.303083 0.0392089i
\(552\) 0 0
\(553\) 7.29150 12.6293i 0.310066 0.537050i
\(554\) 0 0
\(555\) 0.771243 + 1.33583i 0.0327375 + 0.0567029i
\(556\) 0 0
\(557\) −2.70850 4.69126i −0.114763 0.198775i 0.802922 0.596084i \(-0.203277\pi\)
−0.917685 + 0.397309i \(0.869944\pi\)
\(558\) 0 0
\(559\) −22.5830 −0.955159
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.0627 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(564\) 0 0
\(565\) 12.8229 22.2099i 0.539462 0.934376i
\(566\) 0 0
\(567\) −9.11438 + 15.7866i −0.382768 + 0.662973i
\(568\) 0 0
\(569\) 6.58301 0.275974 0.137987 0.990434i \(-0.455937\pi\)
0.137987 + 0.990434i \(0.455937\pi\)
\(570\) 0 0
\(571\) 7.81176 0.326912 0.163456 0.986551i \(-0.447736\pi\)
0.163456 + 0.986551i \(0.447736\pi\)
\(572\) 0 0
\(573\) 8.70850 15.0836i 0.363803 0.630125i
\(574\) 0 0
\(575\) 1.88562 3.26599i 0.0786359 0.136201i
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) −19.2915 33.4139i −0.801727 1.38863i
\(580\) 0 0
\(581\) −28.9373 −1.20052
\(582\) 0 0
\(583\) 29.2288 + 50.6257i 1.21053 + 2.09670i
\(584\) 0 0
\(585\) 6.58301 + 11.4021i 0.272174 + 0.471419i
\(586\) 0 0
\(587\) 7.06275 12.2330i 0.291511 0.504911i −0.682656 0.730739i \(-0.739175\pi\)
0.974167 + 0.225828i \(0.0725088\pi\)
\(588\) 0 0
\(589\) 9.46863 22.7148i 0.390148 0.935946i
\(590\) 0 0
\(591\) −10.1144 + 17.5186i −0.416050 + 0.720620i
\(592\) 0 0
\(593\) −14.8542 25.7283i −0.609991 1.05654i −0.991241 0.132063i \(-0.957840\pi\)
0.381250 0.924472i \(-0.375494\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 52.5830 2.15208
\(598\) 0 0
\(599\) −8.46863 14.6681i −0.346019 0.599322i 0.639520 0.768775i \(-0.279133\pi\)
−0.985538 + 0.169453i \(0.945800\pi\)
\(600\) 0 0
\(601\) 10.4170 0.424918 0.212459 0.977170i \(-0.431853\pi\)
0.212459 + 0.977170i \(0.431853\pi\)
\(602\) 0 0
\(603\) −1.29150 + 2.23695i −0.0525941 + 0.0910956i
\(604\) 0 0
\(605\) −8.70850 + 15.0836i −0.354051 + 0.613234i
\(606\) 0 0
\(607\) −6.93725 −0.281574 −0.140787 0.990040i \(-0.544963\pi\)
−0.140787 + 0.990040i \(0.544963\pi\)
\(608\) 0 0
\(609\) 15.8745 0.643268
\(610\) 0 0
\(611\) −4.35425 + 7.54178i −0.176154 + 0.305108i
\(612\) 0 0
\(613\) 3.70850 6.42331i 0.149785 0.259435i −0.781363 0.624077i \(-0.785475\pi\)
0.931148 + 0.364642i \(0.118809\pi\)
\(614\) 0 0
\(615\) −1.26927 −0.0511821
\(616\) 0 0
\(617\) 15.4373 + 26.7381i 0.621480 + 1.07644i 0.989210 + 0.146503i \(0.0468018\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(618\) 0 0
\(619\) −8.45751 −0.339936 −0.169968 0.985450i \(-0.554366\pi\)
−0.169968 + 0.985450i \(0.554366\pi\)
\(620\) 0 0
\(621\) −2.17712 3.77089i −0.0873650 0.151321i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.14575 7.18065i 0.165830 0.287226i
\(626\) 0 0
\(627\) −32.5203 42.5790i −1.29873 1.70044i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −12.4059 21.4876i −0.493870 0.855408i 0.506105 0.862472i \(-0.331085\pi\)
−0.999975 + 0.00706354i \(0.997752\pi\)
\(632\) 0 0
\(633\) 17.5830 + 30.4547i 0.698862 + 1.21046i
\(634\) 0 0
\(635\) 21.8745 0.868063
\(636\) 0 0
\(637\) 6.29150 + 10.8972i 0.249278 + 0.431763i
\(638\) 0 0
\(639\) 10.8340 0.428586
\(640\) 0 0
\(641\) −6.43725 + 11.1497i −0.254256 + 0.440385i −0.964693 0.263376i \(-0.915164\pi\)
0.710437 + 0.703761i \(0.248497\pi\)
\(642\) 0 0
\(643\) 3.26013 5.64671i 0.128567 0.222685i −0.794555 0.607193i \(-0.792296\pi\)
0.923122 + 0.384508i \(0.125629\pi\)
\(644\) 0 0
\(645\) 49.1660 1.93591
\(646\) 0 0
\(647\) 22.4575 0.882896 0.441448 0.897287i \(-0.354465\pi\)
0.441448 + 0.897287i \(0.354465\pi\)
\(648\) 0 0
\(649\) −18.4373 + 31.9343i −0.723726 + 1.25353i
\(650\) 0 0
\(651\) 27.2288 47.1616i 1.06718 1.84841i
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) −1.59412 2.76110i −0.0622874 0.107885i
\(656\) 0 0
\(657\) 6.83399 0.266619
\(658\) 0 0
\(659\) 9.29150 + 16.0934i 0.361946 + 0.626908i 0.988281 0.152646i \(-0.0487793\pi\)
−0.626335 + 0.779554i \(0.715446\pi\)
\(660\) 0 0
\(661\) 8.11438 + 14.0545i 0.315613 + 0.546657i 0.979568 0.201115i \(-0.0644566\pi\)
−0.663955 + 0.747773i \(0.731123\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −15.8745 20.7846i −0.615587 0.805993i
\(666\) 0 0
\(667\) 1.35425 2.34563i 0.0524367 0.0908231i
\(668\) 0 0
\(669\) −24.8856 43.1032i −0.962134 1.66646i
\(670\) 0 0
\(671\) −2.17712 3.77089i −0.0840470 0.145574i
\(672\) 0 0
\(673\) −13.8745 −0.534823 −0.267411 0.963582i \(-0.586168\pi\)
−0.267411 + 0.963582i \(0.586168\pi\)
\(674\) 0 0
\(675\) 3.03137 + 5.25049i 0.116678 + 0.202092i
\(676\) 0 0
\(677\) 11.4170 0.438791 0.219395 0.975636i \(-0.429592\pi\)
0.219395 + 0.975636i \(0.429592\pi\)
\(678\) 0 0
\(679\) −6.76013 + 11.7089i −0.259430 + 0.449346i
\(680\) 0 0
\(681\) 9.72876 16.8507i 0.372807 0.645720i
\(682\) 0 0
\(683\) −5.41699 −0.207276 −0.103638 0.994615i \(-0.533048\pi\)
−0.103638 + 0.994615i \(0.533048\pi\)
\(684\) 0 0
\(685\) 25.6458 0.979874
\(686\) 0 0
\(687\) 26.4575 45.8258i 1.00942 1.74836i
\(688\) 0 0
\(689\) 12.5830 21.7944i 0.479374 0.830301i
\(690\) 0 0
\(691\) 2.58301 0.0982622 0.0491311 0.998792i \(-0.484355\pi\)
0.0491311 + 0.998792i \(0.484355\pi\)
\(692\) 0 0
\(693\) −33.8745 58.6724i −1.28679 2.22878i
\(694\) 0 0
\(695\) 30.6863 1.16400
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −24.9686 43.2469i −0.944400 1.63575i
\(700\) 0 0
\(701\) 5.17712 8.96704i 0.195537 0.338681i −0.751539 0.659688i \(-0.770688\pi\)
0.947077 + 0.321008i \(0.104022\pi\)
\(702\) 0 0
\(703\) −0.594119 + 1.42526i −0.0224076 + 0.0537548i
\(704\) 0 0
\(705\) 9.47974 16.4194i 0.357028 0.618390i
\(706\) 0 0
\(707\) 24.8745 + 43.0839i 0.935502 + 1.62034i
\(708\) 0 0
\(709\) 1.82288 + 3.15731i 0.0684595 + 0.118575i 0.898223 0.439539i \(-0.144858\pi\)
−0.829764 + 0.558115i \(0.811525\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 0 0
\(713\) −4.64575 8.04668i −0.173985 0.301350i
\(714\) 0 0
\(715\) 15.2915 0.571870
\(716\) 0 0
\(717\) −15.8745 + 27.4955i −0.592844 + 1.02684i
\(718\) 0 0
\(719\) 1.35425 2.34563i 0.0505050 0.0874771i −0.839668 0.543100i \(-0.817250\pi\)
0.890173 + 0.455623i \(0.150584\pi\)
\(720\) 0 0
\(721\) −48.4575 −1.80465
\(722\) 0 0
\(723\) −20.0627 −0.746142
\(724\) 0 0
\(725\) −1.88562 + 3.26599i −0.0700302 + 0.121296i
\(726\) 0 0
\(727\) 0.708497 1.22715i 0.0262767 0.0455126i −0.852588 0.522584i \(-0.824968\pi\)
0.878865 + 0.477071i \(0.158302\pi\)
\(728\) 0 0
\(729\) −41.0000 −1.51852
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 16.1033 0.594788 0.297394 0.954755i \(-0.403883\pi\)
0.297394 + 0.954755i \(0.403883\pi\)
\(734\) 0 0
\(735\) −13.6974 23.7246i −0.505236 0.875094i
\(736\) 0 0
\(737\) 1.50000 + 2.59808i 0.0552532 + 0.0957014i
\(738\) 0 0
\(739\) 16.9059 29.2818i 0.621893 1.07715i −0.367240 0.930126i \(-0.619697\pi\)
0.989133 0.147024i \(-0.0469694\pi\)
\(740\) 0 0
\(741\) −8.87451 + 21.2895i −0.326013 + 0.782090i
\(742\) 0 0
\(743\) −23.7601 + 41.1538i −0.871675 + 1.50978i −0.0114112 + 0.999935i \(0.503632\pi\)
−0.860263 + 0.509850i \(0.829701\pi\)
\(744\) 0 0
\(745\) 9.00000 + 15.5885i 0.329734 + 0.571117i
\(746\) 0 0
\(747\) −15.8745 27.4955i −0.580818 1.00601i
\(748\) 0 0
\(749\) −55.7490 −2.03702
\(750\) 0 0
\(751\) −3.93725 6.81952i −0.143672 0.248848i 0.785204 0.619237i \(-0.212558\pi\)
−0.928877 + 0.370389i \(0.879225\pi\)
\(752\) 0 0
\(753\) 77.3320 2.81814
\(754\) 0 0
\(755\) 10.6458 18.4390i 0.387439 0.671063i
\(756\) 0 0
\(757\) −2.29150 + 3.96900i −0.0832861 + 0.144256i −0.904660 0.426135i \(-0.859875\pi\)
0.821374 + 0.570391i \(0.193208\pi\)
\(758\) 0 0
\(759\) −20.2288 −0.734257
\(760\) 0 0
\(761\) 42.8745 1.55420 0.777100 0.629377i \(-0.216690\pi\)
0.777100 + 0.629377i \(0.216690\pi\)
\(762\) 0 0
\(763\) −26.5830 + 46.0431i −0.962369 + 1.66687i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.8745 0.573195
\(768\) 0 0
\(769\) −17.6458 30.5633i −0.636322 1.10214i −0.986233 0.165359i \(-0.947122\pi\)
0.349911 0.936783i \(-0.386212\pi\)
\(770\) 0 0
\(771\) 32.5203 1.17119
\(772\) 0 0
\(773\) −2.46863 4.27579i −0.0887903 0.153789i 0.818210 0.574920i \(-0.194967\pi\)
−0.907000 + 0.421130i \(0.861633\pi\)
\(774\) 0 0
\(775\) 6.46863 + 11.2040i 0.232360 + 0.402459i
\(776\) 0 0
\(777\) −1.70850 + 2.95920i −0.0612920 + 0.106161i
\(778\) 0 0
\(779\) −0.771243 1.00979i −0.0276327 0.0361796i
\(780\) 0 0
\(781\) 6.29150 10.8972i 0.225128 0.389933i
\(782\) 0 0
\(783\) 2.17712 + 3.77089i 0.0778041 + 0.134761i
\(784\) 0 0
\(785\) −8.70850 15.0836i −0.310820 0.538355i
\(786\) 0 0
\(787\) −42.5203 −1.51568 −0.757842 0.652438i \(-0.773746\pi\)
−0.757842 + 0.652438i \(0.773746\pi\)
\(788\) 0 0
\(789\) −14.4686 25.0604i −0.515097 0.892174i
\(790\) 0 0
\(791\) 56.8118 2.01999
\(792\) 0 0
\(793\) −0.937254 + 1.62337i −0.0332829 + 0.0576476i
\(794\) 0 0
\(795\) −27.3948 + 47.4491i −0.971592 + 1.68285i
\(796\) 0 0
\(797\) 44.8118 1.58731 0.793657 0.608365i \(-0.208174\pi\)
0.793657 + 0.608365i \(0.208174\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.96863 6.87386i 0.140050 0.242573i
\(804\) 0 0
\(805\) −9.87451 −0.348031
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) 0 0
\(811\) 15.6458 + 27.0992i 0.549397 + 0.951583i 0.998316 + 0.0580106i \(0.0184757\pi\)
−0.448919 + 0.893572i \(0.648191\pi\)
\(812\) 0 0
\(813\) 16.3431 + 28.3071i 0.573179 + 0.992775i
\(814\) 0 0
\(815\) −3.23987 + 5.61162i −0.113488 + 0.196566i
\(816\) 0 0
\(817\) 29.8745 + 39.1149i 1.04518 + 1.36846i
\(818\) 0 0
\(819\) −14.5830 + 25.2585i −0.509571 + 0.882604i
\(820\) 0 0
\(821\) −3.00000 5.19615i −0.104701 0.181347i 0.808915 0.587925i \(-0.200055\pi\)
−0.913616 + 0.406578i \(0.866722\pi\)
\(822\) 0 0
\(823\) −15.9373 27.6041i −0.555538 0.962220i −0.997861 0.0653641i \(-0.979179\pi\)
0.442324 0.896855i \(-0.354154\pi\)
\(824\) 0 0
\(825\) 28.1660 0.980615
\(826\) 0 0
\(827\) 26.3229 + 45.5926i 0.915336 + 1.58541i 0.806408 + 0.591359i \(0.201408\pi\)
0.108928 + 0.994050i \(0.465258\pi\)
\(828\) 0 0
\(829\) 17.1660 0.596200 0.298100 0.954535i \(-0.403647\pi\)
0.298100 + 0.954535i \(0.403647\pi\)
\(830\) 0 0
\(831\) −36.4059 + 63.0568i −1.26291 + 2.18742i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 19.7490 0.683443
\(836\) 0 0
\(837\) 14.9373 0.516307
\(838\) 0 0
\(839\) −20.7601 + 35.9576i −0.716719 + 1.24139i 0.245573 + 0.969378i \(0.421024\pi\)
−0.962293 + 0.272016i \(0.912310\pi\)
\(840\) 0 0
\(841\) 13.1458 22.7691i 0.453302 0.785142i
\(842\) 0 0
\(843\) 67.3542 2.31980
\(844\) 0 0
\(845\) 7.40588 + 12.8274i 0.254770 + 0.441275i
\(846\) 0 0
\(847\) −38.5830 −1.32573
\(848\) 0 0
\(849\) −40.5405 70.2182i −1.39135 2.40988i
\(850\) 0 0
\(851\) 0.291503 + 0.504897i 0.00999258 + 0.0173077i
\(852\) 0 0
\(853\) 4.29150 7.43310i 0.146938 0.254505i −0.783156 0.621825i \(-0.786391\pi\)
0.930094 + 0.367321i \(0.119725\pi\)
\(854\) 0 0
\(855\) 11.0405 26.4857i 0.377578 0.905790i
\(856\) 0 0
\(857\) −10.5000 + 18.1865i −0.358673 + 0.621240i −0.987739 0.156112i \(-0.950104\pi\)
0.629066 + 0.777352i \(0.283437\pi\)
\(858\) 0 0
\(859\) −6.61438 11.4564i −0.225680 0.390889i 0.730843 0.682545i \(-0.239127\pi\)
−0.956523 + 0.291656i \(0.905794\pi\)
\(860\) 0 0
\(861\) −1.40588 2.43506i −0.0479123 0.0829865i
\(862\) 0 0
\(863\) 31.0627 1.05739 0.528694 0.848812i \(-0.322682\pi\)
0.528694 + 0.848812i \(0.322682\pi\)
\(864\) 0 0
\(865\) −4.93725 8.55157i −0.167872 0.290762i
\(866\) 0 0
\(867\) −44.9778 −1.52753
\(868\) 0 0
\(869\) 9.29150 16.0934i 0.315193 0.545930i
\(870\) 0 0
\(871\) 0.645751 1.11847i 0.0218804 0.0378980i
\(872\) 0 0
\(873\) −14.8340 −0.502054
\(874\) 0 0
\(875\) 43.7490 1.47899
\(876\) 0 0
\(877\) −20.8229 + 36.0663i −0.703139 + 1.21787i 0.264221 + 0.964462i \(0.414885\pi\)
−0.967359 + 0.253409i \(0.918448\pi\)
\(878\) 0 0
\(879\) 38.2804 66.3036i 1.29117 2.23636i
\(880\) 0 0
\(881\) −36.8745 −1.24233 −0.621167 0.783678i \(-0.713341\pi\)
−0.621167 + 0.783678i \(0.713341\pi\)
\(882\) 0 0
\(883\) 14.1974 + 24.5906i 0.477780 + 0.827539i 0.999676 0.0254701i \(-0.00810828\pi\)
−0.521896 + 0.853009i \(0.674775\pi\)
\(884\) 0 0
\(885\) −34.5608 −1.16175
\(886\) 0 0
\(887\) −8.70850 15.0836i −0.292403 0.506456i 0.681975 0.731376i \(-0.261121\pi\)
−0.974377 + 0.224919i \(0.927788\pi\)
\(888\) 0 0
\(889\) 24.2288 + 41.9654i 0.812606 + 1.40748i
\(890\) 0 0
\(891\) −11.6144 + 20.1167i −0.389096 + 0.673935i
\(892\) 0 0
\(893\) 18.8229 2.43506i 0.629884 0.0814861i
\(894\) 0 0
\(895\) −3.34313 + 5.79048i −0.111749 + 0.193554i
\(896\) 0 0
\(897\) 4.35425 + 7.54178i 0.145384 + 0.251813i
\(898\) 0 0
\(899\) 4.64575 + 8.04668i 0.154944 + 0.268372i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 54.4575 + 94.3232i 1.81223 + 3.13888i
\(904\) 0 0
\(905\) −36.5830 −1.21606
\(906\) 0 0
\(907\) −19.9686 + 34.5867i −0.663047 + 1.14843i 0.316763 + 0.948505i \(0.397404\pi\)
−0.979811 + 0.199927i \(0.935929\pi\)
\(908\) 0 0
\(909\) −27.2915 + 47.2703i −0.905202 + 1.56786i
\(910\) 0 0
\(911\) −16.9373 −0.561156 −0.280578 0.959831i \(-0.590526\pi\)
−0.280578 + 0.959831i \(0.590526\pi\)
\(912\) 0 0
\(913\) −36.8745 −1.22037
\(914\) 0 0
\(915\) 2.04052 3.53428i 0.0674575 0.116840i
\(916\) 0 0
\(917\) 3.53137 6.11652i 0.116616 0.201985i
\(918\) 0 0
\(919\) 19.8745 0.655600 0.327800 0.944747i \(-0.393693\pi\)
0.327800 + 0.944747i \(0.393693\pi\)
\(920\) 0 0
\(921\) −0.854249 1.47960i −0.0281485 0.0487545i
\(922\) 0 0
\(923\) −5.41699 −0.178303
\(924\) 0 0
\(925\) −0.405881 0.703006i −0.0133453 0.0231147i
\(926\) 0 0
\(927\) −26.5830 46.0431i −0.873100 1.51225i
\(928\) 0 0
\(929\) 4.79150 8.29913i 0.157204 0.272285i −0.776655 0.629926i \(-0.783085\pi\)
0.933859 + 0.357640i \(0.116419\pi\)
\(930\) 0 0
\(931\) 10.5516 25.3128i 0.345816 0.829595i
\(932\) 0 0
\(933\) 18.0516 31.2663i 0.590984 1.02361i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −3.56275 6.17086i −0.116390 0.201593i 0.801945 0.597398i \(-0.203799\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(938\) 0 0
\(939\) 23.4797 0.766232
\(940\) 0 0
\(941\) 8.41699 + 14.5787i 0.274386 + 0.475251i 0.969980 0.243184i \(-0.0781920\pi\)
−0.695594 + 0.718435i \(0.744859\pi\)
\(942\) 0 0
\(943\) −0.479741 −0.0156225
\(944\) 0 0
\(945\) 7.93725 13.7477i 0.258199 0.447214i
\(946\) 0 0
\(947\) 3.87451 6.71084i 0.125905 0.218073i −0.796182 0.605058i \(-0.793150\pi\)
0.922086 + 0.386985i \(0.126483\pi\)
\(948\) 0 0
\(949\) −3.41699 −0.110920
\(950\) 0 0
\(951\) 15.8745 0.514766
\(952\) 0 0
\(953\) 6.72876 11.6545i 0.217966 0.377528i −0.736220 0.676742i \(-0.763391\pi\)
0.954186 + 0.299214i \(0.0967245\pi\)
\(954\) 0 0
\(955\) 5.41699 9.38251i 0.175290 0.303611i
\(956\) 0 0
\(957\) 20.2288 0.653903
\(958\) 0 0
\(959\) 28.4059 + 49.2004i 0.917274 + 1.58876i
\(960\) 0 0
\(961\) 0.874508 0.0282099
\(962\) 0 0
\(963\) −30.5830 52.9713i −0.985524 1.70698i
\(964\) 0 0
\(965\) −12.0000 20.7846i −0.386294 0.669080i
\(966\) 0 0
\(967\) −6.64575 + 11.5108i −0.213713 + 0.370162i −0.952874 0.303367i \(-0.901889\pi\)
0.739161 + 0.673529i \(0.235222\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.1974 47.1073i 0.872806 1.51174i 0.0137234 0.999906i \(-0.495632\pi\)
0.859082 0.511838i \(-0.171035\pi\)
\(972\) 0 0
\(973\) 33.9889 + 58.8705i 1.08963 + 1.88730i
\(974\) 0 0
\(975\) −6.06275 10.5010i −0.194163 0.336301i
\(976\) 0 0
\(977\) −7.45751 −0.238587 −0.119293 0.992859i \(-0.538063\pi\)
−0.119293 + 0.992859i \(0.538063\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −58.3320 −1.86240
\(982\) 0 0
\(983\) 15.8745 27.4955i 0.506318 0.876969i −0.493655 0.869658i \(-0.664339\pi\)
0.999973 0.00731102i \(-0.00232719\pi\)
\(984\) 0 0
\(985\) −6.29150 + 10.8972i −0.200464 + 0.347214i
\(986\) 0 0
\(987\) 42.0000 1.33687
\(988\) 0 0
\(989\) 18.5830 0.590905
\(990\) 0 0
\(991\) −1.41699 + 2.45431i −0.0450123 + 0.0779636i −0.887654 0.460511i \(-0.847666\pi\)
0.842641 + 0.538475i \(0.180999\pi\)
\(992\) 0 0
\(993\) −26.2085 + 45.3944i −0.831702 + 1.44055i
\(994\) 0 0
\(995\) 32.7085 1.03693
\(996\) 0 0
\(997\) 8.11438 + 14.0545i 0.256985 + 0.445111i 0.965433 0.260652i \(-0.0839376\pi\)
−0.708448 + 0.705763i \(0.750604\pi\)
\(998\) 0 0
\(999\) −0.937254 −0.0296534
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1216.2.i.k.961.1 4
4.3 odd 2 1216.2.i.l.961.2 4
8.3 odd 2 38.2.c.b.11.1 yes 4
8.5 even 2 304.2.i.e.49.2 4
19.7 even 3 inner 1216.2.i.k.577.1 4
24.5 odd 2 2736.2.s.v.1873.1 4
24.11 even 2 342.2.g.f.163.1 4
40.3 even 4 950.2.j.g.49.3 8
40.19 odd 2 950.2.e.k.201.2 4
40.27 even 4 950.2.j.g.49.2 8
76.7 odd 6 1216.2.i.l.577.2 4
152.3 even 18 722.2.e.o.99.1 12
152.11 odd 6 722.2.a.j.1.2 2
152.27 even 6 722.2.a.g.1.1 2
152.35 odd 18 722.2.e.n.99.2 12
152.43 odd 18 722.2.e.n.389.2 12
152.45 even 6 304.2.i.e.273.2 4
152.51 even 18 722.2.e.o.415.2 12
152.59 even 18 722.2.e.o.595.2 12
152.67 even 18 722.2.e.o.245.1 12
152.75 even 2 722.2.c.j.429.2 4
152.83 odd 6 38.2.c.b.7.1 4
152.91 even 18 722.2.e.o.423.1 12
152.99 odd 18 722.2.e.n.423.2 12
152.107 even 6 722.2.c.j.653.2 4
152.123 odd 18 722.2.e.n.245.2 12
152.125 even 6 5776.2.a.ba.1.1 2
152.131 odd 18 722.2.e.n.595.1 12
152.139 odd 18 722.2.e.n.415.1 12
152.141 odd 6 5776.2.a.z.1.2 2
152.147 even 18 722.2.e.o.389.1 12
456.11 even 6 6498.2.a.ba.1.2 2
456.83 even 6 342.2.g.f.235.1 4
456.179 odd 6 6498.2.a.bg.1.2 2
456.197 odd 6 2736.2.s.v.577.1 4
760.83 even 12 950.2.j.g.349.2 8
760.387 even 12 950.2.j.g.349.3 8
760.539 odd 6 950.2.e.k.501.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.c.b.7.1 4 152.83 odd 6
38.2.c.b.11.1 yes 4 8.3 odd 2
304.2.i.e.49.2 4 8.5 even 2
304.2.i.e.273.2 4 152.45 even 6
342.2.g.f.163.1 4 24.11 even 2
342.2.g.f.235.1 4 456.83 even 6
722.2.a.g.1.1 2 152.27 even 6
722.2.a.j.1.2 2 152.11 odd 6
722.2.c.j.429.2 4 152.75 even 2
722.2.c.j.653.2 4 152.107 even 6
722.2.e.n.99.2 12 152.35 odd 18
722.2.e.n.245.2 12 152.123 odd 18
722.2.e.n.389.2 12 152.43 odd 18
722.2.e.n.415.1 12 152.139 odd 18
722.2.e.n.423.2 12 152.99 odd 18
722.2.e.n.595.1 12 152.131 odd 18
722.2.e.o.99.1 12 152.3 even 18
722.2.e.o.245.1 12 152.67 even 18
722.2.e.o.389.1 12 152.147 even 18
722.2.e.o.415.2 12 152.51 even 18
722.2.e.o.423.1 12 152.91 even 18
722.2.e.o.595.2 12 152.59 even 18
950.2.e.k.201.2 4 40.19 odd 2
950.2.e.k.501.2 4 760.539 odd 6
950.2.j.g.49.2 8 40.27 even 4
950.2.j.g.49.3 8 40.3 even 4
950.2.j.g.349.2 8 760.83 even 12
950.2.j.g.349.3 8 760.387 even 12
1216.2.i.k.577.1 4 19.7 even 3 inner
1216.2.i.k.961.1 4 1.1 even 1 trivial
1216.2.i.l.577.2 4 76.7 odd 6
1216.2.i.l.961.2 4 4.3 odd 2
2736.2.s.v.577.1 4 456.197 odd 6
2736.2.s.v.1873.1 4 24.5 odd 2
5776.2.a.z.1.2 2 152.141 odd 6
5776.2.a.ba.1.1 2 152.125 even 6
6498.2.a.ba.1.2 2 456.11 even 6
6498.2.a.bg.1.2 2 456.179 odd 6